1. Field of the Invention
The present invention generally relates to method, system and computer program product used in design and analysis of a structure, more particularly to practical fast mesh-free analysis of a general three-dimensional structure.
2. Description of the Related Art
Finite element analysis (FEA) is a computerized method widely used in industry to model and solve engineering problems relating to complex systems since its invention in late 1950's. With the advent of the modern digital computer, FEA has been implemented as FEA computer program product. Basically, the FEA computer program product is provided with a model of the geometric description and the associated material properties at certain points within the model. In this model, the geometry of the system under analysis is represented by solids, shells and beams of various sizes, which are called elements. The vertices of the elements are referred to as nodes. The individual elements are connected together by a topological map, which is usually called mesh. The model is comprised of a finite number of elements, which are assigned a material name to associate with material properties. The model thus represents the physical space occupied by the object under analysis along with its immediate surroundings. The FEA computer program product then refers to a table in which the properties (e.g., stress-strain constitutive equation, Young's modulus, Poisson's ratio, thermo-conductivity) of each material type are tabulated. Additionally, the conditions at the boundary of the object (i.e., loadings, physical constraints, etc.) are specified. In this fashion a model of the object and its environment is created.
Although FEA has been successfully applied to many fields to simulate various engineering problems, there are some instances that FEA may not be advantageous due to numerical compatibility condition is not the same as the physical compatibility condition of a continuum. For example, in Lagrangian type of computations, one may experience mesh distortion, which can either end the computation altogether or result in dramatic deterioration of accuracy. In addition, the FEA often requires a very fine mesh in problems with high gradients or a distinct local character, which can be computationally expensive. For this reason, adaptive FEA has been developed.
Adaptive re-meshing procedures for simulations of impact/penetration problems, explosion/fragmentation problems, flow pass obstacles, and fluid-structure interaction problems etc., have become formidable tasks to undertake. The difficulties here are not only re-meshing, but also mapping the state variables from the old mesh to the new mesh. Hence, this process often introduces numerical errors; frequent re-mesh is not desirable. Another procedure called Arbitrary Lagrangian Eulerian (ALE) formulations has been developed. Its objective is to move the mesh independently of the material so that the mesh distortion can be minimized. Unfortunately, the mesh distortion still creates severe numerical errors for very large strain and high speed mechanical simulations especially when the mesh is coarse. An example is the strain localization problem, which is notorious for its mesh alignment sensitivity. Therefore it is computationally efficacious to discretize a continuum by a set of nodal points without mesh constraints.
The mesh-free analysis has become one of the focused research topics during the 1990's. Many applications of using mesh-free analysis have been achieved in the past decade. In comparison with conventional finite element methods, the characteristics of smoothness and naturally conforming of the approximation, exemption from meshing, and higher convergence rate and the easy of nodal insertion and deletion have make mesh-free methods attractive alternative numerical techniques for nonlinear analysis of industrial applications. The main disadvantage of mesh-free method against its popularity is the high computational requirement (i.e., high CPU cost). The high CPU cost is primarily resulting from the usage of high order of integration rule, the introduction of more neighboring information, the imposition of essential boundary conditions and the computation of the variable transfer. Recently, several recent advances have been made to enhance the computational efficiency. However, the improvements are still limited and most of the methods are not robust in the nonlinear and large deformation region. The major reasons are the inaccurate spatial integration and the lack of interpolation property in the mesh-free approximation. Therefore, it would be desirable to have a fast and practical mesh-free method for the general industrial applications with desired accuracy, robustness and wide applicability.